ssSARIMA¶

Purpose¶

Estimates SARIMA models using a state space representation, the Kalman filter, and maximum likelihood.

Format¶

sOut = ssSARIMA(y, p, d, q, p_s, d_s, s[, const, trend])¶

Parameters:

y (Nx1 vector) – data.
p (Scalar) – the autoregressive order.
d (Scalar) – Optional argument, the order of differencing. Default = 0.
q (Scalar) – Optional argument, the moving average order. Default = 0.
const (Scalar) – Optional argument, an indicator variable to include a constant in the model. Set to 1 to include trend, 0 otherwise. Default = 0.
trend (Scalar) – Optional argument, an indicator variable to include a trend in the model. Set to 1 to include trend, 0 otherwise. Default = 0.

Returns:

sOut (Struct) –

an instance of an ssOut structure. For an instance named sOut, the members are:

sOut.final_params

String array, final parameter estimates.

sOut.resid

Vector, residuals.

sOut.fitted

Vector, the fitted y values based on final parameter estimates.

sOut.df_model

Vector, degrees of freedom of the model.

sOut.df_resid

Vector, degrees of freedom of the residuals.

sOut.numObs

Vector, number of observations.

sOut.mleResults

Instance of a cmlmtResults structure. Further information provided in the cmlmt documentation.

sOut.kfResults

Instance of a kalmanOut structure, contains the results from the kalmanFilter().

kfResults.filtered_state	Matrix, k_endog x numObs, filtered states.
kfResults.filtered_state_cov	Array, numObs x k_endog x k_endog, filtered state covariances.
kfResults.predicted_state	Matrix, k_endog x (numObs+1), predicted states.
kfResults.predicted_state_cov	Array, numObs x k_endog x k_endog, predicted state covariances.
kfResults.forecast	Matrix, k_endog x numObs, forecasts.
kfResults.forecast_error	Matrix, k_endog x numObs, forecast error.
kfResults.forecast_error_cov	Array, numObs x k_endog x k_endog, forecast error covariances.
kfResults.loglikelihood	Matrix, k_endog x (numObs+1), computed loglikelihood.

sOut.ssmFinal

Instance of a ssModel structure, contains the final state space system matrices used in the kalmanFilter(). Contains the following members:

ssmFinal.Z	k_endog x k_states, transition matrix.
ssmFinal.d	k_endog x 1, observation intercept.
ssmFinal.H	k_endog x k_endog, observation disturbance covariance.
ssmFinal.T	k_states x k_states, design matrix.
ssmFinal.c	k_states x k_states, state intercept.
ssmFinal.R	k_states x k_posdef, selection matrix.
ssmFinal.Q	k_states x k_posdef, state disturbance covariance.
ssmFinal.a_0	k_states x 1, initial prior state mean.
ssmFinal.p_0	k_states x k_states, initial prior state covariance.

sOut.aic

Scalar, model Akaike’s information criterion.

sOut.aicc

Scalar, model corrected Akaike’s information criterion.

sOut.bic

Scalar, model Schwarz’ Bayesian information criterion.

sOut.hqic

Scalar, model Hannan–Quinn information criterion.

sOut.ssy

Scalar, sum of squares total (Deviations of y from mean of y).

sOut.sse

Scalar, sum of squared errors.

sOut.mse

Scalar, means squared errors.

sOut.rsquared

Scalar, model r-squared.

sOut.ljung_box

Scalar, Ljung-Box Q-test for autocorrelation.

sOut.ljung_box_pval

Scalar, p-value of the Ljung-Box Q-test for autocorrelation.

sOut.hetero_test

Scalar, tests for the null hypothesis of no heteroskedasticity.

sOut.hetero_test_pval

Scalar, p-value of the test for the null hypothesis of no heteroskedasticity.

sOut.jb_stat

Scalar, the Jarque-Bera goodness-of-fit test on model residuals.

sOut.jb_stat_pval

Scalar, p-value ofthe Jarque-Bera goodness-of-fit test.

sOut.standardized_forecast_errors

Scalar, standardized forecast errors used in all residual diagnostics.

sOut.skew

Scalar, sample skewness of the standardized forecast errors.

sOut.kurtosis

Scalar, sample kurtosis of the standardized forecast errors.

sOut.irf

Scalar, model impulse response functions.

sOut.forecasts

Scalar, forecasts.

Example¶

 new;
 library tsmt, cmlmt, sslib;

 // Create file name with full path
 fname = __FILE_DIR $+ "wpi1.dat";

 // Load variable 'wpi' from 'wpi1.dat'
 y = loadd(fname, "wpi");

// Model settings
p = 1;
d = 1;
q = 1;
trend = 0;
const = 1;

// Declare 'sOut' to be an ssOut structure
// to hold the estimation results and then
// estimate the model
struct ssOut sOut;
sOut = ssARIMA(y, p, d, q, const, trend);

The example above prints the following results

 Return Code:                                                             0
 Log-likelihood:                                                 -4.603e+04
 Number of Cases:                                                       130
 AIC:                                                             9.207e+04
 AICC:                                                            9.207e+04
 BIC:                                                             9.208e+04
 HQIC:                                                            9.207e+04
 Covariance Method:                                    ML covariance matrix
==========================================================================

       Parameters         Estimates         Std. Err.            T-stat             Prob.          Gradient
-------------------------------------------------------------------------------------------------------------
   theta : e[t-1]           -0.4073            0.0753           -5.4103            0.0000                 .
  theta : e[t-12]           -0.5506            0.0891           -6.1764            0.0000                 .
           Sigma2            0.0014            0.0002            9.2080            0.0000                 .

 Correlation matrix of the parameters
--------------------------------------------------------------------------

     1.0000     -0.2894     -0.0638
    -0.2894      1.0000     -0.0066
    -0.0638     -0.0066      1.0000

 Model and residual diagnostics:
==========================================================================

 Ljung-Box (Q):                                                      0.0381
 Prob(Q):                                                             0.845
 Heteroskedasticity (H):                                              0.544
 Prob(H):                                                             0.049
 Jarque-Bera (JB):                                                     2.02
 Prob(JB):                                                            0.363
 Skew:                                                               0.0671
 Kurtosis:                                                             3.59
==========================================================================

Source¶

ssarima.src